Best Known (189−21, 189, s)-Nets in Base 3
(189−21, 189, 159436)-Net over F3 — Constructive and digital
Digital (168, 189, 159436)-net over F3, using
- net defined by OOA [i] based on linear OOA(3189, 159436, F3, 21, 21) (dual of [(159436, 21), 3347967, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3189, 1594361, F3, 21) (dual of [1594361, 1594172, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3189, 1594368, F3, 21) (dual of [1594368, 1594179, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3189, 1594368, F3, 21) (dual of [1594368, 1594179, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3189, 1594361, F3, 21) (dual of [1594361, 1594172, 22]-code), using
(189−21, 189, 531456)-Net over F3 — Digital
Digital (168, 189, 531456)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3189, 531456, F3, 3, 21) (dual of [(531456, 3), 1594179, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3189, 1594368, F3, 21) (dual of [1594368, 1594179, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3189, 1594368, F3, 21) (dual of [1594368, 1594179, 22]-code), using
(189−21, 189, large)-Net in Base 3 — Upper bound on s
There is no (168, 189, large)-net in base 3, because
- 19 times m-reduction [i] would yield (168, 170, large)-net in base 3, but